Kerodon

Warning 8.6.5.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is both a cartesian fibration and a cocartesian fibration. Then $U$ admits a both a cocartesian dual $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ and a cartesian dual $U'': \operatorname{\mathcal{E}}'' \rightarrow \operatorname{\mathcal{C}}$. For every vertex $C \in \operatorname{\mathcal{C}}$, there are equivalences of $\infty $-categories $\operatorname{\mathcal{E}}'_{C} \simeq \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \simeq \operatorname{\mathcal{E}}''_{C}$. Beware that the fibrations $U'$ and $U''$ are generally not equivalent to one another (see Example 8.6.5.5).